Geometry

Special Right Triangles

Big Picture

Some triangles, called special right triangles, have simple formulas for calculating their side lengths. These include isosceles right triangles and triangles with the measures of 30°, 60°, and 90°. By memorizing these special right triangles, we can solve certain geometry problems faster.

Key Terms

Isosceles Triangle: A triangle with at least two sides of equal length.

Isosceles Right Triangle: An isosceles triangle with the three angle measures 45°, 45°, and 90°.

Equilateral Triangle: A triangle where all three sides are of equal length.

Isosceles Right Triangle

An isosceles right triangle can be formed by cutting a square diagonally in half. All isosceles right triangles have the three angle measures 45°, 45°, and 90°. These triangles are also called a 45-45-90 triangle.

Since the three angles are always the same, all isosceles right triangles are similar.

Isosceles Right Triangle

If the variable  to represent the length of the leg, the Pythagorean theorem can be used to find the hypotenuse. The side lengths of an isosceles right triangle always form an extended ratio of x:x:x2.

Theorem: In a 45°-45°-90° triangle, the hypotenuse is 2 times as long as each leg.

  • hypotenuse = leg • 2

30-60-90 Triangle Theorem

A 30-60-90 triangle has angle measures of 30°, 60°, and 90°.

  • Since the three angles are always the same, all isosceles right triangles are similar.
30-60-90 Triangle

All the angle measures in an equilateral triangle are 60°, so dividing an equilateral triangle in half will create two 30-60-90 triangles.

30-60-90 Triangle Theorem

Theorem: In a 30-60-90 triangle, the side lengths of the will always form an extended ratio of x : x3 : 2x

  • hypotenuse = 2 • shorter leg
  • longer leg = shorter leg • 3

So the altitude of the equilateral triangle is \(\frac{\sqrt{3}}{2}\) times the side length of the equilateral triangle.

Helpful Hint: If you forget these relationships, the Pythagorean Theorem can still be used to find the  missing side length.

Geometry

Special Right Triangles Cont.

Trigonometric Ratios for Special Right Triangles

Using the side length relationships for special right triangles, we can find the trigonometric ratios for 30°, 45°, and 60° angles.

For 30°:

  • \(\sin 30^{\circ} = \frac{x}{2x} = \frac{1}{2}\)
  • \(\sin 30^{\circ} = \frac{\sqrt{3}x}{2x} = \frac{\sqrt{3}}{2}\)
  • \(\sin 30^{\circ} = \frac{x}{\sqrt{3}x} = \frac{\sqrt{3}}{3}\)

For 45°:

  • \(\sin 45^{\circ} = \frac{x}{\sqrt{2}x} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
  • \(\sin 45^{\circ} = \frac{x}{\sqrt{2}x} = \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}}\)
  • \(\tan 45^{\circ} = \frac{x}{x} = 1\)

For 60°:

  • \(\sin 60^{\circ} = \frac{\sqrt{3}x}{2x} = \frac{\sqrt{3}}{2}\)
  • \(\sin 60^{\circ} = \frac{x}{2x} = \frac{1}{2}\)
  • \(\tan 60^{\circ} = \frac{\sqrt{3}x}{x} = \sqrt{3}\)

Notes